
We adapt the Euclidean projection algorithm in [1]}. We have \(x_i > 0\)  is implicitly enforced by the negentropy mirror map \(\Phi \) , because \(\Phi \)  is locally increasing at the boundary of \(\mathcal {D}\) , the minimizer can't lie in a point that has a component \(x_i =0\)  for any \(i\in \mathcal {N}\) . \(D_{\Phi } (\mathbf {x},\mathbf {y})\)  is convex in \(\mathbf {x}\) , \(D_{\Phi } (\mathbf {x},\mathbf {y}) = \Phi (\mathbf {x}) - \Phi (\mathbf {y}) - \Phi (\mathbf {y})^T (\mathbf {x} - \mathbf {y})\) , since \(\Phi (\mathbf {x})\)  is strongly convex [2]} over \(\mathcal {X}\)  and \(-\Phi (\mathbf {y})^T (\mathbf {x} - \mathbf {y})\)  is linear in \(\mathbf {x}\) , then we can find a unique solution to the above optimization problem. The Lagrangian function of the above problem:
\(\mathcal {L} (\mathbf {x}, \mathbf {\beta }, \gamma ) = \sum ^N_{i=1} \left(x_i\mathrm {log}\left(\frac{x_i}{y_i}\right)- x_i + y_i\right) - \sum ^N_{i=1} \beta _i \left(1-x\right) - \gamma \left(\sum ^N_{i=1} x_i - k\right)\) 
